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Article

Heat Transfer Behaviors in Horizontal WellsConsidering the Effects of Drill Pipe Rotation, andHydraulic and Mechanical Frictions duringDrilling Procedures

Xin Chang 1,2, Jun Zhou 1,2,*, Yintong Guo 1,2, Shiming He 3,*, Lei Wang 1,2, Yulin Chen 4,Ming Tang 3 and Rui Jian 5

1 State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics,Chinese Academy of Science, Wuhan 430071, Hubei, China; [email protected] (X.C.);[email protected] (Y.G.); [email protected] (L.W.)

2 University of Chinese Academy of Sciences, Beijing 100049, China3 State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University,

Chengdu 610500, Sichuan, China; [email protected] PetroChina Co. Ltd., Chuandong Drilling Company, Chongqing 401147, China; [email protected] CNOOC Ltd., Zhanjiang Branch, Zhanjiang 524057, Guangdong, China; [email protected]* Correspondence: [email protected] (J.Z.); [email protected] (S.H.);

Tel.: +86-184-283-01012 (J.Z.); +86-139-818-94760 (S.H.)

Received: 30 August 2018; Accepted: 10 September 2018; Published: 12 September 2018��

Abstract: Horizontal wells are increasingly being utilized in the exploration and development of oiland gas resources. However, the high temperature that occurs during drilling processes leads to anumber of problems, such as the deterioration of drilling fluid properties and borehole instability.Therefore, the insight into heat transfer behaviors in horizontal wells is certainly advantageous.This study presents an integrated numerical model for predicting the temperature distribution duringhorizontal wells drilling considering the effects of drill pipe rotations, and hydraulic (i.e., circulatingpressure losses) and mechanical frictions. A full implicit finite difference method was applied to solvethis model. The results revealed that the mechanical frictions affect more on wellbore temperaturevariation than the effects of heat transfer intensification and circulating pressure losses; Moreover,the drilling fluid temperature was found higher than the stratum temperature at horizontal section,the temperature difference at the bottom hole reached up to 16 ◦C if pressure drops, heat transferstrengthened by rotations and mechanical frictions were all taken into account. This research couldbe utilized as a theoretical reference for predicting temperature distributions and estimating risks inhorizontal wells drilling.

Keywords: heat transfer; wellbore temperature; drill pipe rotation; friction; horizontal well

1. Introduction

The circulating temperature in the wellbore during the drilling procedure is a matter worthexamining. The circulating temperature inevitably affects the properties of operating fluids duringthe drilling of oil–gas wells, and temperature fluctuations in the wellbore and formations expose thesurrounding rocks to the risk of collapse. Moreover, temperature impacts the creep rates of soft rockinterlayers, such as salt, gypsum, and mudstone. In addition, the corrosion of downhole tools andexpansion of gas flowing with the drilling fluid are known to be related to the temperature profile.All these aspects mentioned above significantly affect well safety during drilling.

Energies 2018, 11, 2414; doi:10.3390/en11092414 www.mdpi.com/journal/energies

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It is generally difficult to predict the downhole temperature profile during the drilling procedurebecause of the complex and volatile drilling conditions. In previous decades, several studies wereconducted on wellbore temperature. Edwardson [1] presented a method to approximately predictwellbore and formation temperature. Ramey et al. [2] introduced a time function, f (t), and total heattransfer coefficient, and computed the circulating temperature considering fluid injection. Their studylaid the foundation for further research on wellbore temperature. Holmes and Swift [3], and Arnold [4]deduced an analytical model for estimating wellbore temperature under a steady state. Hasan andKabir [5], inspired by the law of pore pressure diffusion in formations, categorized the thermalconduction in formations as radial, one-dimensional, and steady-state heat transfers, and usedthe inversion of the Laplace transformation to understand the temperature distribution in rocksaround wellbores. Furthermore, they modified the time function f (t), which was subsequently widelyapplied [6,7]. Based on the studies of Ramey [2], and Holmes and Swift [3], Kabir and Hasan [7]established a temperature model for reverse circulation. Aniket et al. [8], and Kumar and Samuel [9]focused on extended-reach drilling and casing while drilling, respectively. Moreover, they introducedan analytical model, including a friction heat source between the drill pipe and wellbore wall,into the model of Kabir and Hasan. The study of Livescu and Wang [10] provided an insight on thewellbore temperature during coiled tubing drilling. The aforementioned investigations were mainlyconducted on the basis of analytical methods. With the development of computational technology,further numerical methods were introduced to calculate the wellbore temperature during the drillingprocedure. Raymond [11] derived energy equations for the fluid in the drill pipe and annulus, as wellas in surrounding rocks by utilizing the finite element method to obtain the numerical solution of thetemperature profile under steady and pseudo-steady states. Latterly, Keller [12] provided detaileddiscrete forms of energy equations, taking into consideration integral wellbore geometric structures,including cement sheaths and casing pipes. Marshall and Bentsen [13] applied a different formula todetermine the effects of circulating pressure losses on temperature in the drill pipe, annulus, and drillbit. The study assumed turbulent flow in the drill pipe and laminar flow in the annulus. The drillingfluid was treated as a power-law fluid. Recent research continued to focus on temperature distributionduring well drilling. Song and Guan [14] proposed a temperature–pressure coupling model foraerated underbalanced drilling in deep water. Yang [15] probed the wellbore-stratum temperature,which encountered the loss of drilling fluid. In addition to the finite element method mentioned above,other solving algorithms were still applied, such as the finite volume method [16,17], artificial neuralnetwork [18], and semi-analytical method [19].

For the temperature profiles of horizontal wells, Siu and Li [20], Yoshioka and Zhu [21,22], andMuradov and Davies [23] examined the temperature distribution of horizontal segments when fluidsflowed into or from formations. For the drilling procedure, Trichel and Fabian [24] investigatedhigh-temperature horizontal wells and discussed energy loss during drilling. A field engineeringsoftware was used to calculate torque, drag force, and hydraulic friction. However, certain equationsand properties of the drilling fluid were not clear. Nguyen and Miska [25], who made derivationsbased on the investigation conducted by Kabir and Hasan on the temperature of a vertical well andconsidered the mechanical friction between drill pipes and wellbore wall, employed an analyticalmodel to predict the circulating temperature of horizontal wells, but ignored the effect of hydraulicfriction. The work completed by Li and Liu [17] on the temperature of horizontal wells includedinformation on the heat source of viscous dissipation, which was similar to the hydraulic frictionin drilling fluids and other mechanical frictions. Nevertheless, the study would have been morecomprehensive if the effect of the key parameter, Nu, was considered. Therefore, for horizontal wells itis advantageous to delve into the integrated impact of all heat sources induced by wellbore geometryand drilling fluid circulation.

In this study, a comprehensive numerical model intended for estimating wellbore and formationtemperature distribution during horizontal well drilling was established according to the law ofconservation of energy, and a five-point finite difference scheme was employed in the solution process.

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The drilling fluid was treated as a non-Newtonian yield-power-law fluid, which accorded with thebehavior of drilling fluid used at the site. Moreover, the effects of drill pipe eccentricity and drillpipe rotation on the heat transfer procedure in horizontal wells were taken into account. Accordingly,not only were conventional operational parameters related to wellbore temperature, for instance,circulating time, displacement, and geothermal gradient, considered, but also other factors thatinfluenced mechanical and hydraulic frictions, such as heat transfer efficiency, drill pipe eccentricityand rotation. This study was aimed at providing an accurate prediction of temperature distributionsin wellbores and formations during horizontal well drilling.

2. Wellbore Geometry and Heat Transfer Mathematical Model

2.1. Description of Wellbore Geometry

Figure 1 shows the wellbore geometry of a horizontal well, mainly including the drill pipe, columnspace in the drill pipe, annulus space between the outer drill pipe wall and wellbore wall, drill bit,and surrounding rock. The drilling fluid flowed down from ground surface within the drill pipe andthrough the drill bit, and thereafter flowed up along the annulus space, and eventually, was back to theground surface. The heat transfer, in the form of conduction within solid parts and convection withinfluid zones, the same as that between fluids and solids, always occurred during the whole process ofcirculating and drilling. The practical wellbore geometry was very complex, and the size and shapeof the wellbore and annulus were affected by the motion, deformation and rotation of drill pipes.Moreover, other factors, such as lithological characters, drilling rate, and weight on the bit also madedifferences. Therefore, the trajectory and shape of the annulus were not as regular as those shown inFigure 1.

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drill pipe rotation on the heat transfer procedure in horizontal wells were taken into account. Accordingly, not only were conventional operational parameters related to wellbore temperature, for instance, circulating time, displacement, and geothermal gradient, considered, but also other factors that influenced mechanical and hydraulic frictions, such as heat transfer efficiency, drill pipe eccentricity and rotation. This study was aimed at providing an accurate prediction of temperature distributions in wellbores and formations during horizontal well drilling.

2. Wellbore Geometry and Heat Transfer Mathematical Model

2.1. Description of Wellbore Geometry

Figure 1 shows the wellbore geometry of a horizontal well, mainly including the drill pipe, column space in the drill pipe, annulus space between the outer drill pipe wall and wellbore wall, drill bit, and surrounding rock. The drilling fluid flowed down from ground surface within the drill pipe and through the drill bit, and thereafter flowed up along the annulus space, and eventually, was back to the ground surface. The heat transfer, in the form of conduction within solid parts and convection within fluid zones, the same as that between fluids and solids, always occurred during the whole process of circulating and drilling. The practical wellbore geometry was very complex, and the size and shape of the wellbore and annulus were affected by the motion, deformation and rotation of drill pipes. Moreover, other factors, such as lithological characters, drilling rate, and weight on the bit also made differences. Therefore, the trajectory and shape of the annulus were not as regular as those shown in Figure 1.

Figure 1. Sketch of wellbore geometry.

Some assumptions for our model require introduction, as follows:

(1) For horizontal wells, the drill pipes in vertical segments were either concentric or eccentric at a constant eccentricity. Built‐up segments were linearly and increasingly eccentric along the wellbore axis, with the eccentricity increasing from e = 0, at the kick‐off point, to e = 1, at the heel of the horizontal segment; the mean eccentricity was e = 0.62. The horizontal segments were totally eccentric (e = 1).

(2) No buckling segments existed along the entire well, and the shapes of drill pipes and annuli were regular.

(3) The effect of casing pipes was not considered as its high heat conductivity. Moreover, the effects of heat radiation and conduction in the drilling fluid along the depth direction were neglected, merely considering the main forms of heat transfer conduction in solid parts and convection in fluid zones.

(4) The initial geothermal temperature in the formation along the depth linearly increased. Based on the above assumptions, the heat transfer model was established according to the law of

r1r2r3

Figure 1. Sketch of wellbore geometry.

Some assumptions for our model require introduction, as follows:

(1) For horizontal wells, the drill pipes in vertical segments were either concentric or eccentric ata constant eccentricity. Built-up segments were linearly and increasingly eccentric along thewellbore axis, with the eccentricity increasing from e = 0, at the kick-off point, to e = 1, at the heelof the horizontal segment; the mean eccentricity was e = 0.62. The horizontal segments weretotally eccentric (e = 1).

(2) No buckling segments existed along the entire well, and the shapes of drill pipes and annuliwere regular.

(3) The effect of casing pipes was not considered as its high heat conductivity. Moreover, the effectsof heat radiation and conduction in the drilling fluid along the depth direction were neglected,merely considering the main forms of heat transfer conduction in solid parts and convection influid zones.

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(4) The initial geothermal temperature in the formation along the depth linearly increased. Based onthe above assumptions, the heat transfer model was established according to the law ofthermodynamic energy conservation. Accordingly, they were used to estimate the temperaturevariation during the drilling procedure.

2.2. Mathematical Model of Heat Transfer

According to works [13,15,17], the first three assumptions enumerated above primarily pertainedto the temperature of vertical wells. Although Li [17] analyzed the temperature of horizontal wells,the investigation did not include the effect of hydraulic friction, which was different from what weconducted in this study. As described in Section 2.1, heat transfer not only occurred within solidparts and fluid zones, but also transpired at interfaces between solids and fluids. In this section,the segregation of a small segment, dz, along the depth direction in order to derive the mathematicalheat transfer model for each part is discussed.

2.2.1. Thermal Equilibrium Equations

(1) Drilling Fluid in drill pipes:

− clρlq∂Tli∂z

+ αdi•2πr1(Td − Tli) + Qi = clρlπr12∂Tli∂t

(1)

In the above, cl is the specific heat of the drilling fluid; ρl is the drilling fluid density; q is thecirculating volume flow rate; r1 is the inner radius of the drill pipe; Tli and Td are the temperature of thedrilling fluid in the drill pipe and annulus, respectively; αdi is the convective heat-transfer coefficientat the inner face of the drill pipe; Qi is the heat source from the drill pipe; z is the well depth; t is thecirculating time.

(2) Drill pipe:

λd∂2Td∂z2

+2r1αdi

r22 − r12(Tli − Td)−

2r2αdor22 − r12

(Td − Tla) = cdρd∂Td∂t

(2)

Here, λd and cd are the heat conductivity coefficient and specific heat of the drill pipe wall,respectively; ρd is the drill pipe density; Td is the temperature of the drill pipe, and Tla is the drillingfluid temperature in the annulus; αdi and αdo are the convective heat-transfer coefficients at the innerand outer faces, respectively.

(3) Drilling Fluid in annular sections:

clρlq∂Tla∂z

+ αdo•2πr3(Tb − Tla)− αb•2πr2(Tla − Td) + Qo = clρl(

r32 − r22)∂Tla

∂t(3)

In this equation, Tb is the temperature at the wellbore wall; αb is the convective heat-transfercoefficient at the wellbore wall; Qo is the heat source in the annulus; r3 is the outer radius of the annulus.

(4) Formation:∂Tr∂t

= α

(∂2Tr∂r2

+1r

∂Tr∂r

+∂2Tr∂z2

)(4)

In the above expression, α = λr/(Crρr); Tr is the stratum temperature; r is the radius from thewellbore axis to a certain point in the stratum; λr is the heat conductivity coefficient of the rock; Cr isthe specific heat of the rock; ρr is the rock density.

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2.2.2. Boundary and Initial Conditions

Boundary and initial conditions were required for solving the heat transfer model. Because theground surface temperature was not difficult to determine, the initial temperature in the formation,which increased linearly, was determined as well. The inlet temperature of the drilling fluid at thewell head was similarly detectable. The temperature at a certain distance from the wellbore axis wasalways undisturbed and retained the initial state. The temperature at the borehole bottom was equalto that in the drilling fluid in the drill pipe and annulus, and drill pipe wall. In addition, an importantcoupled boundary, which was the quantity of heat that went through the formation into the boreholeby heat conduction, was equal to that of the drilling fluid in the annulus, absorbed from the wellborewall by heat convection.

(1) Boundary conditions:

• Inlet condition: Tli|z=0,t=t = Tin;• Temperatures at the wellbore bottom: Tli|z=H,t=t = Td|z=H,t=t = Tla|z=H,t=t ;• Coupled conditions at the wellbore wall: αb(Tr − Tla)|r=rb = −λr

(∂Tr∂r

)∣∣∣r=rb

;

• Far field boundary: Tr = Tsur f + GT · H ( r → ∞ );

(2) Initial conditions:

• Stratum retaining the original temperature: Tr = Tsur f + GT · H;• Drilling fluid in the drill pipe: Tli = Tin + G f · H;• Drilling fluid in the annulus: Tla = Tout + G f · H;

The heat exchange model mentioned above was solved by the finite difference method, and thewellbore-stratum temperature field was determined after iteration. It was worth noting that the heatsources, i.e., the heat generated by the circulating pressure drop and mechanical friction between thedrill pipe and wellbore wall, were considered in this model. However, before the model was solved,it was discretized. Details are provided in Appendix A.1.

2.3. Calculation of Heat Transfer Coefficients

2.3.1. Methods of Determining Heat Transfer Coefficients in Previous Studies

The convective heat transfer coefficient was generally determined by a dimensionless number,the Nusselt number, Nu. For the heat transfer in the drill pipe, Nu, differed from that in the annulus.Different types of experimental methods were designed to determine Nu, whose value varied underdifferent experiments. As for forced convective heat transfer, Nu was related to two other dimensionlessnumbers—Reynolds number, Re, and Prandtl number, Pr, and can be expressed as Nu = f (Re,Pr).These numbers can be computed using the following equations: Nu = αDH/λ, Re = ρvDH/µ, Pr = µCp/λ.In this paper, the methods for determining Nu were reviewed.

(1) Determination of Nu in the drill pipe

Laminar Regime:Nu = 4.36 or 4.364 f or qs = constantNu = 3.66 f or Ts = constant

(5)

Some scholars [26–31] utilized the first unchanged Nu (Nu = 4.364) when they investigated thelaminar flow in drill pipes with a constant qs (i.e., qs = constant), as shown in Equation (5). Keller [12],and Mashall and Bentsen [13] selected a smaller value, i.e., Nu = 4.12. Moreover, Petersen [32] andLi [17] used a coefficient that was dependent on the flow pattern index of the drilling fluid. Specifically,

Nu = 4.364((3n + 1)/4n)0.323 (6)

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Turbulent Regime:

Dittus and Boelter [33] introduced Equation (7) as a simplified model for calculating Nu inturbulent pipe flow:

Nu = 0.023 Re4/5Prn

(n = 0.4 f or heating, 0.3 f or cooling)(7)

Equation (7) is applicable when Re > 104, 0.7 ≤ Pr ≤ 160, and L/D ≥ 10. Moreover, if Re wascomputed using DH, then this would be the same as the formula utilized by Santoyo-Gutierrez [26] toestimate Nu in the annulus flow under the turbulent regime. This expression had been widely usedbecause of its brevity.

Based on the expression of Dittus and Boelter, Seider and Tate [34] took into consideration thevariation in fluid properties. Thus, the term (µ/µs)0.14 to correlate the expression of Dittus and Boelterwas introduced:

Nu = 0.027Re4/5Pr1/3(µ/µs)0.14

(Re > 104 ; 0.7 ≤ Pr ≤ 16700 ; L/D ≥ 10)(8)

The above formula is simple but not more precise than that what Petukhov [35] proposed, which isEquation (9) below:

Nu = ( f /8)RePr1.07+12.7( f /8)1/2(Pr2/3−1)

104 < Re < 5× 106; 0.5 < Pr < 200(9)

where f is the friction coefficient, which can refer to that of the Moody plate. For a smooth tube,f = [1.82log(Re) − 1.64]−2. The equation provided by Petukhov was appropriate only when the flowwas fully turbulent, i.e., when Re > 104. Subsequently, Gnielinski [36] applied it to the transitional flowregime (2300 < Re < 104):

Nu = ( f /8)(Re−1000)Pr1+12.7( f /8)1/2(Pr2/3−1)

2300 < Re < 5× 106; 0.5 < Pr < 200

Smooth Pipe : f = (0.79lnRe− 1.64)−2(10)

Gnielinski [36,37] also introduced the entrance correction factor, 1 + (D/L)2/3, to correlate theeffect of the entrance side, as well as another factor, (Pr/Prw)0.11, to correlate with the variation in fluidproperty. The factor f obtained by various studies differed, such as f = [1.8log(Re) − 1.5]−2 [29,37].Yang [31], used the formula of Gnielinski with another expression of factor f, and obtained theexpression below:

f = a/Reb (a = (log(n) + 3.93)/50; b = (1.75− log(n))/7) (11)

Transitional Regime

As for Nu under the transitional regime, the same formula for turbulent flow was used by some todetermine number. However, the most popular expression was the correlated equation recommendedby Gnielinski [26,27,30,31], although there were others who employed linear interpolation to calculateNu [29,38]:

Nu = (1− γ)Nucrit,lam + γNucrit,turbγ =

Re−Recrit,LamRecrit,Turb−Recrit,Lam (1 ≤ γ ≤ 1)

(12)

where Nucrit,Lam and Recrit,Lam are the critical Nusselt and Reynolds numbers from the laminar regime totransitional regime, respectively. Similarly, Recrit,Turb and Nucrit,Turb are the critical Nusselt and criticalReynolds numbers from the transitional regime to turbulent regime.

(2) Determination of Nu in the Annulus

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Laminar Regime

In drill engineering, two methods were used to determine Nu in the annulus. The first was thesame as the laminar flow drill pipe Nu, which was equal to 4.364 [30] or 4.12 [12,13]; the second wasthe widely used expression formulated by Seder and Tate [17,26,31,39], with the factor µ/µw set to 1when the change in fluid property was not considered:

Nu = 1.86(RePr)13(

DHL

) 13(

µµw

)0.14Re < 2300; 0.48 < Pr < 16700; RePr(DH/L) > 10

(13)

Turbulent Regime

In early research, Nu for turbulent flow in the annulus was generally calculated by means of theformula of Dittus and Boelter [33] or that of Seider and Tate [34] with the factor (µ/µs)0.14. In laterstudies, the expression deduced by Gnielinski or some other derivative equations were applied.Gnielinski [37] presented another expression to consider the effects of the inlet side, variation in thefluid property, and boundary of the annulus:

Nu =( f /8)RePr

k + 12.7√

f /8(Pr23 − 1)

[1 +

(dhL

) 23](

PrPrw

)0.11Fann (14)

where:

k = 1.07 + 900Re −0.63

1+10Pr ; f = [1.8log(Re)− 1.5]−2;

Fann = 0.75a−0.17 f or outer wall insulated; Fann = 0.9− 0.15a0.6 f or inner wall insulated

Li [17] proposed another expression for Nu in the annulus:

Nu = ARee f f αPrγ (15)

where Reeff is the effective Reynolds number, considering drill pipe rotation. Specific values for A, α,and γ were not available.

Transitional Regime

There were three methods for the calculation of Nu for transitional flow in the annulus. The firstused the expression of Dittus and Boelter [12], the second used the correlations of Gnielinski [31],and the third was by linear interpolation [29].

2.3.2. Heat Transfer Coefficients Involving Drill-Pipe Rotation

In this discussion, only the effect of rotation on the heat transfer in the annulus was considered,whereas that in the drill pipe was neglected. When the rotational speed exceeded a certain value,Taylor vortices emerged, and according to studies [40–42], four flow modes can occur and be classified,as shown in Figure 2.

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Figure 2. Different flow modes in annulus with inner cylinder rotating.

Under these four flow modes of varying degrees, the rotational speed of drill pipes and axial flow velocity in the annulus affected the heat transfer efficiency. By means of the dimensionless Taylor number, Ta, which was relevant to the rotation and axial Reynolds number, Rea, and depending on the axial flow velocity, the flow mode can be determined. In previous studies, the flow mode in the annulus was treated either as a purely laminar flow or turbulent flow, and ignored the effect of drill pipe rotation. The equation provided by Seider and Tata [34] was merely recommended for the purely turbulent phase, and was used by Keller [12], and Tragesser and Crawford [39] to calculate the temperature profile during the drilling procedure.

In terms of the heat transfer in the annulus geometry, two dimensionless numbers should be introduced in the axial Reynolds number (Rea = ρVaDH/μ) and Taylor number (Ta = ρ2ω2Ri(DH/2)3/μ2), where DH = 2(Ro − Ri) is the hydraulic diameter, and Ri and Ro are the inner and outer radii of the annulus, respectively. The two most common means employed to analyze the heat transfer in a rotating inner pipe included the conduct of various experimental investigations and numerical stimulations. There are generally four forms in estimating Nu, as follows:

(a) effNu ARe Pr (b) aNu ARe Pr Ta

(c) effNu ARe (d) Nu ATa

where the constants A, α, β, γ depend on experimental conditions; Reeff is the effective Reynolds

number defined as Reeff = ρVeffDH/μ with 22eff axialV v R 22

eff axialV v R

22eff axialV v R given by Gazley [43]. In previous investigations on wellbore temperature, the influence of the drill pipe rotation on

heat transfer was seldom mentioned [12,13,19,31,44]. In this study, focus was set on this aspect. Becker [40] demonstrated the heat transfer process in the annulus in detail. On the other hand, G.I.Taylor [45] researched on the spiral flow in the annulus, and combined theory with the experimental method. Taylor showed that the laminar flow became unstable flow and a type of secondary flow, called Taylor vortex, emerged. Taylor defined the critical rotary speed beyond which Taylor vortices will occur:

4 223 22i o

ci

r rPb r

(16)

where

1

0.0571 1 0.652 0.00056 1 0.652i i

b bPr r

align.

Taylor stated that when b/ri = 0, a dimensionless set of numbers, 2 3 2 1697c mr b , determined whether the Taylor vortex will occur. This dimensionless group was named Taylor number.

Taylor Number (Ta)

Axial Reyno

lds Num

ber (Re

a) Pure Turbulent Flow

Pure LaminarFlow

Laminar + Taylor VorticesFlow

Turbulent + Taylor VorticesFlow

Figure 2. Different flow modes in annulus with inner cylinder rotating.

Under these four flow modes of varying degrees, the rotational speed of drill pipes and axialflow velocity in the annulus affected the heat transfer efficiency. By means of the dimensionless Taylornumber, Ta, which was relevant to the rotation and axial Reynolds number, Rea, and depending onthe axial flow velocity, the flow mode can be determined. In previous studies, the flow mode inthe annulus was treated either as a purely laminar flow or turbulent flow, and ignored the effect ofdrill pipe rotation. The equation provided by Seider and Tata [34] was merely recommended for thepurely turbulent phase, and was used by Keller [12], and Tragesser and Crawford [39] to calculate thetemperature profile during the drilling procedure.

In terms of the heat transfer in the annulus geometry, two dimensionless numbers should beintroduced in the axial Reynolds number (Rea = ρVaDH/µ) and Taylor number (Ta = ρ2ω2Ri(DH/2)3/µ2),where DH = 2(Ro − Ri) is the hydraulic diameter, and Ri and Ro are the inner and outer radii of theannulus, respectively. The two most common means employed to analyze the heat transfer in a rotatinginner pipe included the conduct of various experimental investigations and numerical stimulations.There are generally four forms in estimating Nu, as follows:

(a) Nu = AReαe f f Prβ (b) Nu = AReαa Pr

βTaγ (c) Nu = AReαe f f (d) Nu = ATaγ

where the constants A, α, β, γ depend on experimental conditions; Reeff is the effective Reynolds

number defined as Reeff = ρVeffDH/µ with Ve f f =√

v2axial + α(ωR)2Ve f f =

√v2axial + α(ωR)

2Ve f f =√v2axial + α(ωR)

2 given by Gazley [43].In previous investigations on wellbore temperature, the influence of the drill pipe rotation on heat

transfer was seldom mentioned [12,13,19,31,44]. In this study, focus was set on this aspect. Becker [40]demonstrated the heat transfer process in the annulus in detail. On the other hand, G.I.Taylor [45]researched on the spiral flow in the annulus, and combined theory with the experimental method.Taylor showed that the laminar flow became unstable flow and a type of secondary flow, called Taylorvortex, emerged. Taylor defined the critical rotary speed beyond which Taylor vortices will occur:

ω2c =π4ν2(ri + ro)

2Pb3r2i(16)

where P = 0.0571[1− 0.652 bri

]+ 0.00056

[1− 0.652 bri

]−1align.

Taylor stated that when b/ri = 0, a dimensionless set of numbers, ω2c rmb3/ν2 = 1697, determinedwhether the Taylor vortex will occur. This dimensionless group was named Taylor number.Chandrasekhar [46] gave a new value for the Taylor number, i.e., Tac = 1708, under the same

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experimental conditions. Goldstein [47] provided a similar value, Tac = 1714. The values of Tacmentioned above were obtained by setting the axial velocity equal to zero. Because the axial velocitycan weaken the transmission of Taylor vortices, it contributed to the stability of the laminar state.The critical Taylor number increased with increasing axial velocity [40], and according to Goldstein [47],when the axial Reynolds number increased from 0 to 31, Tac increased to 1966. Kaye [41] introduced aTaylor number, Ta, which was different from that presented by Chandrasekhar [46]. By considering theeffect of the annulus shape and using a geometrical factor, Fg, Ta can be expressed as shown below:

Ta∗ =ωr

12mb

32

ν(17)

Fg = π2

41.2

(1− b2rm

)−1P−

12

where P = 0.0571[1− 0.652 b/rm1−b/2rm

]+ 0.00056

[1− 0.652 b/rm1−b/2rm

]−1 (18)The correlated Taylor number, Tam = Ta/Fg, was named modified Taylor number. Kaye [41] defined

another Taylor number, Ta*, and provided the critical value, Tac* = 41.2, under the condition b/rm = 0.The square of Tac* is equal to 1697, which approximated the critical value of Ta provided by Taylor.Thus, Ta = Ta*2. When the formula for Ta stated by Taylor was applied, Fg, had to be replaced by Fg2.Accordingly, the critical Taylor number correlated by the geometrical factor is given as Tac = 1697Fg2 orTac* = 41.2Fg. Becker [40] maintained that the heat transfer efficiency had no relationship with the axialand tangential velocities under the purely laminar flow regime, whereas, the opposite was true whenTaylor vortices occurred in the laminar flow regime. Because the drilling fluid flow was of the laminarstate, only the Nusselt number of a purely laminar regime and laminar plus Taylor vortices regimewere considered here.

Purely laminar flow

For a purely laminar flow, the formula provided by Seder and Tate [34] was widely used despitevariations in the properties of the drilling fluid. Thus, with µ/µw = 1, then:

Nu = 1.86(RePr)13 (DH/L)

13

Re < 2300; 0.48 < Pr < 16700; RePr(DH/L) > 10(19)

Laminar flow plus Taylor vortices

For the flow in the annulus with Taylor vortices under the laminar flow regime, the equations ofSimmers and Coney [48] were used, as follows:

Nu =4PrRe0.5a Ta0.3675

B(

A1−N

)0.5( N1−N

)0.25Ta0.6175c

(20)

where

A =[1 + N2 + (1− N2)/lnN

][2 + (1− N2)/lnN

]−1B = Pr + ln

{1 + Pr× exp

[23

(1−N

N

)0.25( NA(1−N)2

)0.5Re−0.5a Ta0.1325Ta0.1175c − 1

]− Pr

}N = Ri/Ro

where text and outside box.The expression above for Nu can be correlated by the coefficient δ1/3 = ((3n + 1)/4n 1/3, which was

presented by Metzner [49] for non-Newtonian fluids. Incidentally, Aoki and Nohira [50] fitted

Energies 2018, 11, 2414 10 of 28

experimental data using air, water, spindle oil, and isobutyl alcohol as experimental fluids anddefined Nu for the laminar plus Taylor vortices state without axial velocity, as follows:

Nu = 0.22(Ta/Fg

)1/4Pr0.3(5000 < Ta/Fg < 2× 105) (21)As for the flow in the drill pipe, Cannon and Kays [51] explained that when fluid flowed in a

pipe rotating about its axis, the effect of rotation on heat transfer was only detectable when the flowwas under the transitional regime and negligible under the laminar and turbulent regimes. Therefore,it was assumed that there was no difference between the heat transfer in a rotary pipe and that in astable one. For the condition of non-Newtonian fluid flows under the laminar regime, the coefficientδ1/3 = ((3n + 1)/4n)1/3 can be utilized for correlating Nu. Thus, Nu = 4.364δ1/3. The expressions of Nuunder other flow regimes were the same as those where pipe rotation was neglected. Having obtainedthe values of Nu, then, the temperature profiles in the wellbore were determined.

2.4. Calculation of Internal Heat Sources

2.4.1. Heat Sources Induced by Mechanical Friction

(1) Heat source generated by friction at the drill bit

Using different types of drill bits for different categories of formations, the friction at the drill bitvaried as well. Nguyen [25] and Warrant [52] presented the following function to estimate the heatsource generated by friction at the drill bit:

q =1J(1− η)(WOB ∗ ROP + 2πωTbit) (22)

where J is Joule’s constant, which is unity when both sides of Equation (22) are in a consistent unitsystem; η is the drill bit efficiency of the portion of work used for penetration; WOB is the weight onthe bit; ROP is the rate of penetration; ω is rotary speed, RPS; Tbit is the torque on the bit.

(2) Heat source generated by friction between drill pipe and wellbore

Vertical Hole

In drilling in vertical sections, we assumed that the drill pipes do not come into contact with thewellbore wall. Thus, friction between the drill pipe and wellbore wall was zero, and there was no heatsource produced along vertical sections.

Curved Hole (Built-Up or Drop-Off Section)

Aadnøy and Andersen [53] defined the drag forces and torques under drilling conditions invarious well segments (shown in Figure 3), including pulling up and lowering down drill pipes.


experimental data using air, water, spindle oil, and isobutyl alcohol as experimental fluids and defined Nu for the laminar plus Taylor vortices state without axial velocity, as follows:

1/ 4 0.3 50.22 / 5000 / 2 10g gNu Ta F Pr Ta F (21) As for the flow in the drill pipe, Cannon and Kays [51] explained that when fluid flowed in a

pipe rotating about its axis, the effect of rotation on heat transfer was only detectable when the flow was under the transitional regime and negligible under the laminar and turbulent regimes. Therefore, it was assumed that there was no difference between the heat transfer in a rotary pipe and that in a stable one. For the condition of non‐Newtonian fluid flows under the laminar regime, the coefficient δ1/3 = ((3n + 1)/4n)1/3 can be utilized for correlating Nu. Thus, Nu = 4.364δ1/3. The expressions of Nu under other flow regimes were the same as those where pipe rotation was neglected. Having obtained the values of Nu, then, the temperature profiles in the wellbore were determined.

2.4. Calculation of Internal Heat Sources

2.4.1. Heat Sources Induced by Mechanical Friction

(1) Heat source generated by friction at the drill bit

Using different types of drill bits for different categories of formations, the friction at the drill bit varied as well. Nguyen [25] and Warrant [52] presented the following function to estimate the heat source generated by friction at the drill bit:

1 1 * 2 bitq WOB ROP TJ (22)

where J is Joule’s constant, which is unity when both sides of Equation (22) are in a consistent unit system; η is the drill bit efficiency of the portion of work used for penetration; WOB is the weight on the bit; ROP is the rate of penetration; ω is rotary speed, RPS; Tbit is the torque on the bit.

(2) Heat source generated by friction between drill pipe and wellbore

Vertical Hole

In drilling in vertical sections, we assumed that the drill pipes do not come into contact with the wellbore wall. Thus, friction between the drill pipe and wellbore wall was zero, and there was no heat source produced along vertical sections.

Curved Hole (Built‐Up or Drop‐Off Section)

Aadnøy and Andersen [53] defined the drag forces and torques under drilling conditions in various well segments (shown in Figure 3), including pulling up and lowering down drill pipes.

(a) (b) (c)

Figure 3. Forces and geometries of various hole profiles [53]. (a) Straight inclined section; (b) Built‐up section; (c) Drop‐off section.

Figure 3. Forces and geometries of various hole profiles [53]. (a) Straight inclined section; (b) Built-upsection; (c) Drop-off section.

Energies 2018, 11, 2414 11 of 28

(a) Drop-off section When pulling up drill pipes, the force at the upper side of each section was:

F2 = F1eµ(α2−α1) + wR1+µ2[(

1− µ2)(

sinα2 − eµ(α2−α1)sinα1)− 2µ

(cosα2 − eµ(α2−α1)cosα1

)](23)

When lowering down the drill pipe, the force at the upper side was:

F2 = F1e−µ(α2−α1) + wR[sinα2 − e−µ(α2−α1)sinα1

](24)

The torque was given by:

T = µr[(F1 + wRsinα1)|α2 − α1| − 2wR(cosα2 − cosα1)] (25)

(b) Built-up section When pulling up drill pipes:

F2 = F1e−µ(α2−α1) − wR[sinα2 − e−µ(α2−α1)sinα1

](26)

When lowering down drill pipes:

F2 = F1eµ(α2−α1) − wR1+µ2[(

1− µ2)(

sinα2 − eµ(α2−α1)sinα1)− 2µ

(cosα2 − eµ(α2−α1)cosα1

)](27)

The torque was:

T = µr[(F1 + wRsinα1)|α2 − α1|+ 2wR(cosα2 − cosα1)] (28)

(c) Straight inclined hole

For tangential sections, the inclination of the wellbore was constant and drill pipes were laiddown on the low side of the borehole. The drag force at the upper side of each wellbore section was

F2 = F1 + w∆s(cosα± µsinα) (29)

where “+” was for pulling-up processes, and “−” was for lowering-down processes. The torque was

T = µrw∆s · sinα (30)

It was noticeable that the drill pipes were always immerged in drilling fluids; therefore, buoyancyshould be taken into consideration. The parameter w is the buoyed pipe weight and computed as:

w = βwd (31)

where wd is the weight of the drill pipe in air, and β is the buoyancy factor. Aadnoy [54] defined it asfollows: (a) When the drilling fluid density in drill pipes was not equal to that in the annulus,β = 1 − (ρo − Ao − ρiAi)/[ρpipe(Ao − Ai)], where ρ is the density, A is the cross-sectional area.The subscripts o, i, and pipe represented the fluid in the annulus, fluid in the drill pipes, and drill pipesthemselves, respectively; (b) When the drilling fluid density in drill pipes equaled that in the annulus,then:

β = 1− ρo/ρpipe (32)

Energies 2018, 11, 2414 12 of 28

Using Equations (23)–(32), the torque in drill pipes can be calculated because of the frictionbetween the drill pipe and wellbore wall. The heat quantity produced by the drill pipe per unit timecan be computed as well [55], as follows:

Q f riction = 2π · RPS · T (33)

where RPS denotes rotations per second, r/s. Equation (33) was used to estimate the heat source offriction between the drill pipe and wellbore wall. After discretizing the entire drill pipe into a numberof sections along the direction of the well depth, the total heat source can be computed by addingthe heat source at each section—from the first section, at the well bottom, to the last section, at thewell head. Moreover, the friction factor, µ, highly depended on the roughness of contact surfaces,properties of the drilling fluid, and temperature and pressure conditions. Craig [56] analyzed 33 wellsin a Norwegian platform and reported that µ = 0.24 was the most applicable friction factor for alltypes of wells, and the condition of the well trajectory had an insignificant effect on the friction factor.Table 1 lists the friction factors between the drill pipes and wellbore wall using different drilling fluids,as reported by Samuel [57].

Table 1. Range of friction factors [57].

Fluid Type Friction Factors

Cases Hole Open Hole

Oil-based 0.16–0.20 0.17–0.25Water-based 0.25–0.35 0.25–0.40

Brine 0.30–0.4 0.3–0.4Polymer-based 0.15–0.22 0.2–0.3Synthetic-based 0.12–0.18 0.15–0.25

Foam 0.30–0.4 0.35–0.55Air 0.35–0.55 0.40–0.60

2.4.2. Heat Sources of Circulating Pressure Losses in Dill Pipe, Annulus, and Drill Bit

The drilling fluid flowed in drill pipes and the annulus gap between the outer wall of drill pipesand wellbore wall. This resulted in pressure drops because of the friction between the fluid andsolid. This energy was eventually converted to heat sources along the well pipes. Keller [12] appliedan empirical method to estimate heat sources during drilling procedures: 20% of the drilling pumphydraulic power loses in drill pipes, 8.5% loses in the annulus, and 70% loses at the drill bit. Based onthe assumption that flow was turbulent in the drill pipes and drill bit, and laminar flow in the annulusspace, the properties of the drilling fluid were in accord with those of the power-law fluid. Marshalland Bentsen [13] derived the following formulas to calculate circulating pressure drops:

• Drill String:

∆Pd =2 f ρV2L

Dgc1√

f=

4.07n

logr× 103

ε+ 6.0− 2.65

n(34)

• Drill Bit:

∆Pb =ρ

2g

(q

0.95An

)(35)

• Annulus:

∆Pa =2KL

(ro − ri)gc

[2(n + 1)q

nπ(ro + ri)(ro − ri)2

]n(36)

where f is the Fanning friction factor, D is the inner diameter of the drill pipe, gc is a unit conversionfactor (127.094 × 106 m/h2), g is the acceleration of gravity, q is the pump displacement, An is the area

Energies 2018, 11, 2414 13 of 28

of bit nozzles, n is the fluid flow behavior index, K the fluid consistency index (kg/h2−n·m), ro and riare outer and inner diameters of the annulus, respectively, and L is length of the drill pipe and annulus.

In this study, the method of Reed and Pilehvari [58], and Kelessidis [59] for non-Newtonian fluidssubjected to the Herschel-Bulkley model was used to compute pressure drops in the drill pipe andannulus. Moreover, the effects of the rotary drill pipe with its eccentricity on the pressure drop in theannulus were considered.

The above expressions presented by Marshall and Bentsen [13] did not consider the effects ofrotation and eccentricity of the drill pipes. In this study, only the effects of these two on the pressuredrop in the annulus were investigated; however, it supposed that they have no influence on the drillpipe. As for non-Newtonian fluids, Cartalos and Dupuis [60] presented a ratio of the pressure drop inthe annulus with a rotary inner pipe to the pressure drop with a stationary inner pipe, as shown below:

Rrot =(dP/dL)rot

(dP/dL)non−rot=

[1 +

32

(So(z)

d

)2]1/2(37)

where d = ro − ri, So(z) is the amplitude of the inner pipe at a certain depth and replaceable with amean value, Savg. Thus, the mean eccentricity is eavg = Savg/d, which is applicable to the whole well.

Through numerical stimulation, Ooms and Kampman–Reinhartz [61] studied the rotation of theinner pipe in relation to the pressure drop in the annulus when eccentricity was present with the dropduring drilling in a slim borehole. Under a low rotary speed, they used a perturbation calculation toderive a formula for estimating the rotation in relation to pressure drop. The correction coefficient is

Rrot =[1 + 32 e

2]× [1 + 32 e2 + 1739200 δ2Ta2 ε2(2+ε2)2 f (e)]−1

f (e) = −13120 + 38112e2 − 10608e4 − 33062e6 − 1221e8(38)

where δ = Ro − Ri is the annulus space gap, e is the dimensionless eccentricity, e = dc/δ; dc is thedistance between the center points of drillpipe and wellbore, and Ta is the Taylor number based on theouter radius of the annulus, i.e., Ta = ρωRo(Ro − Ri)/µ. They reported that only when δεTa is muchless than 1 can Equation (38) be applied. When the Taylor number was high, numerical simulationshould be employed.

Ahmed [62] presented a fitted formula for the rotation correction coefficient based on experimentaldata derived from the yield-power-law drilling fluid. The coefficient was named PLR, defined asthe ratio of the pressure drop in the annulus with a rotary inner pipe to that in the annulus with astationary pipe:

Rrot = 0.36×(

13.5 +τy

ρV2a

)0.428× e0.158avg × n0.054 × Ta0.0319 × Re0.042e f f × k

(1k− 1)−0.0152

(39)

where Ta = Di(Do−Di)3

16

(ρω

µapp

)2; Ree f f =

8ρV2aτw,lam

; k = DiDo ; µapp =τy.γ+ K

.γ

n−1.

In the study of Anifowoshe and Osisanya [63],.γ =

√.γ

2a +

.γ

2θ =

√(12Va(2n+1)

Dh ·3n

)2+(

ωDiDh

)2,

where τy and n are the fluid yield stress and flow behavior index for Herschel-Bulkley fluid,respectively; Va is the mean velocity of the fluid in annulus; k is the ratio of the inner diameterto the outer diameter of the annulus; µapp is the apparent viscosity;

.γ is the resultant shearing rate,

combining axial and tangential velocities; τw,lam is the mean shear stress on the wall. Ahmed [62]indicated that the above formula was applicable when the mean eccentricity, eavg, was between 0.5 and1.0, the effective Reynolds number, Reeff, ranged from 721 to 2397, and Ta0.5 was between 479 and 1602.

Haciislamoglu and Langlinais [64], and Haciislamoglu [65] introduced two correction coefficientsto take into account the effects of rotary and eccentric drill pipes for calculating the pressure dropwhen yield-power-law drilling fluid flowed through the annulus.

Energies 2018, 11, 2414 14 of 28

1© Laminar regime:

Recc =(dP/dL)ecc(dP/dL)con

= 1− 0.072 en(

DiDo

)0.8454− 1.5e2

√n(

DiDo

)0.1852+ 0.96e3

√n(

DiDo

)0.2527(40)

2© Turbulent regime:

Recc =(dP/dL)ecc(dP/dL)con

= 1− 0.048 en(

DiDo

)0.8454− 23 e2

√n(

DiDo

)0.1852+ 0.285e3

√n(

DiDo

)0.2527(41)

Haciislamoglu and Langlinais [64] stated that the two expressions were valid when 0 ≤ e ≤ 0.95,0.3 ≤ Di/Do ≤ 0.9, 0.4 ≤ n ≤ 1.0. Here, the eccentricity is equal to 0 for vertical well segments(concentric), 0.62emax for inclined well segments (linearly eccentric), and 1 for horizontal well segments(completely eccentric). Pilehvari and Serth [66] modified the above formula for the laminar regime byreplacing e/n with e·n:

Recc = 1− 0.1019en(

DiDo

)−0.4675− 1.6152e2n0.085

(DiDo

)0.7875+ 1.1434e3n0.0547

(DiDo

)1.1655(42)

The application range of n was extended to 0.2 ≤ n ≤ 1.0, the scope of the ratio Di/Do became0.2 ≤ Di/Do ≤ 0.8, and the range of eccentricity, e, was unchanged. Using the modified expression,Recc is 1 as n→ 0, instead of Recc→−∞. All the requirements in the abovementioned new formula canbe satisfied under all practical conditions in drilling operations. If the pressure drop in a concentricannulus was determined, then the correction coefficients can be used to take into account the effectsof rotation and eccentricity of drill pipes, and the effects on pressure drop can finally be reflected indifferent wellbore temperature profiles.

3. Analysis of Wellbore Temperature Behavior during Deep Horizontal Well Drilling

3.1. Influences of Circulating Pressure Drop under Various Rotation Rates and Eccentricities of Drill Pipes

As presented in Section 2.4.2, the pressure drop in the annuli of drill pipes can be computed.Accordingly, it can be transferred to heat sources, which contributed to temperature fluctuations alongthe wellbore. In this study, we assumed that all the heat generated within the drill pipes flowedwith the drilling fluid in the drill pipes, and all the heat produced at the drill bit and annulus wastransferred to the drilling fluid in the annulus. Firstly, the pressure drops in the drill pipe and annulususing the input parameters summarized in Table A1 in this study were calculated.

As shown in Figure 4, both pressure drops in the drill pipe and annulus increased with the increasein flow rate, which meant more heat was produced because of the hydraulic friction between the solidand fluid regions. In addition, when the eccentricity increased, pressure decreased. Furthermore,when an inner drill pipe rotated, the pressure drop in the annulus was higher than that with a stationaryone (notice that in Figure 4b that Ahmed’s fitting expression did not conform to this law because itwas not applicable when the flow rate was low).

Energies 2018, 11, 2414 15 of 28Energies 2018, 11, x FOR PEER REVIEW 15 of 29

Figure 4. The pressure drop (a) in the drill pipe and (b) annulus.

Therefore, it was evident that the eccentricity of the inner drill pipes led to a decline in the pressure drop in the annulus, whereas the increase in rotatory speed produced reversed results. Based on input data in this study, the formula derived by Cartalos and Dupuis [60] was involved in the flow analysis.

According to the calculation procedure shown in Figure A4, the temperature profiles, considering drill pipe rotations, and hydraulic and mechanical frictions were obtainable. In Figure 5a, when the effect of the circulating pressure drop was considered, the temperature along the wellbore was higher than that when the pressure drop was neglected. The temperature difference, ΔT, tended to increase with increasing well depth and reached its maximum at the borehole bottom with the value ΔTa ≈ 3 °C. Moreover, the phenomenon, in which the maximum temperature along the wellbore was found at a certain distance from the borehole bottom, which was mentioned in a previous research on the temperature of vertical wellbores, occurred here as well when the effect of the circulating pressure drop was excluded. On the other hand, when the effect of the pressure drop was considered, the phenomenon disappeared. Meanwhile, temperature spurts occurred at the well bottom because of the substantial amount of hydraulic mechanical energy loss as the drilling fluid flowed through the bit nozzles. Figure 5b compares the temperature profile when only the pressure drop at the drill bit was taken into account and when pressure drop was neglected. However, a temperature difference of ΔTa ≈ 2 °C, which exhibited a high influence of the pressure drop on the drill bit, still remained.

Figure 5. (a) Influence of pressure drops of the whole wellbore on temperature profile (b) Influence of pressure drop of drill bit on temperature profile.

According to Figures 6 and 7, the eccentricity and rotation of drill pipes had insignificant effects on the temperature distribution along wellbore. When the eccentricities for the vertical, inclined, and horizontal sections were set to e = 0.6, e = 0.62, and e = 1, respectively, and the magnitude of revolutions per minute (RPM) of drill pipes was 100 r/min, the temperature difference at the wellbore bottom was only approximately 0.2 °C, which was negligible.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

250

500

750

1000

1250

1500

1750

2000

Flow Rate(m3/s)

Pre

ssur

e G

radi

ent (

Pa/

m)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

200

300

400

500

600

700

800

Flow Rate(m3/s)

Pre

ssur

e G

radi

ent(P

a/m

)

concentric(=0)Haciislamoglu(ecc, =0.6)Ahmed(ecc+rot, =0.6, RPM=60 r/min)Cartalos(ecc+rot, =0.6, RPM=60 r/min)

15 20 25 30 35 40 45 50 550

500

1000

1500

2000

2500

3000

Temperature( )℃

Mea

sure

d D

epth

(m)

WithoutPressure Drop

WithPressure Drop

DrillpipeAnnulusGeothermal

15 20 25 30 35 40 45 50 550

500

1000

1500

2000

2500

3000

Temperature( )℃

Mea

sure

d D

epth

(m)

WithoutBit Pressure Drop

WithBit Pressure Drop


(a) (b)

(a) (b)


Therefore, it was evident that the eccentricity of the inner drill pipes led to a decline in the pressuredrop in the annulus, whereas the increase in rotatory speed produced reversed results. Based on inputdata in this study, the formula derived by Cartalos and Dupuis [60] was involved in the flow analysis.

According to the calculation procedure shown in Figure A4, the temperature profiles, consideringdrill pipe rotations, and hydraulic and mechanical frictions were obtainable. In Figure 5a, when theeffect of the circulating pressure drop was considered, the temperature along the wellbore washigher than that when the pressure drop was neglected. The temperature difference, ∆T, tended toincrease with increasing well depth and reached its maximum at the borehole bottom with the value∆Ta ≈ 3 ◦C. Moreover, the phenomenon, in which the maximum temperature along the wellbore wasfound at a certain distance from the borehole bottom, which was mentioned in a previous researchon the temperature of vertical wellbores, occurred here as well when the effect of the circulatingpressure drop was excluded. On the other hand, when the effect of the pressure drop was considered,the phenomenon disappeared. Meanwhile, temperature spurts occurred at the well bottom because ofthe substantial amount of hydraulic mechanical energy loss as the drilling fluid flowed through thebit nozzles. Figure 5b compares the temperature profile when only the pressure drop at the drill bitwas taken into account and when pressure drop was neglected. However, a temperature difference of∆Ta ≈ 2 ◦C, which exhibited a high influence of the pressure drop on the drill bit, still remained.



Therefore, it was evident that the eccentricity of the inner drill pipes led to a decline in the pressure drop in the annulus, whereas the increase in rotatory speed produced reversed results. Based on input data in this study, the formula derived by Cartalos and Dupuis [60] was involved in the flow analysis.

According to the calculation procedure shown in Figure A4, the temperature profiles, considering drill pipe rotations, and hydraulic and mechanical frictions were obtainable. In Figure 5a, when the effect of the circulating pressure drop was considered, the temperature along the wellbore was higher than that when the pressure drop was neglected. The temperature difference, ΔT, tended to increase with increasing well depth and reached its maximum at the borehole bottom with the value ΔTa ≈ 3 °C. Moreover, the phenomenon, in which the maximum temperature along the wellbore was found at a certain distance from the borehole bottom, which was mentioned in a previous research on the temperature of vertical wellbores, occurred here as well when the effect of the circulating pressure drop was excluded. On the other hand, when the effect of the pressure drop was considered, the phenomenon disappeared. Meanwhile, temperature spurts occurred at the well bottom because of the substantial amount of hydraulic mechanical energy loss as the drilling fluid flowed through the bit nozzles. Figure 5b compares the temperature profile when only the pressure drop at the drill bit was taken into account and when pressure drop was neglected. However, a temperature difference of ΔTa ≈ 2 °C, which exhibited a high influence of the pressure drop on the drill bit, still remained.

Figure 5. (a) Influence of pressure drops of the whole wellbore on temperature profile (b) Influence of pressure drop of drill bit on temperature profile.

According to Figures 6 and 7, the eccentricity and rotation of drill pipes had insignificant effects on the temperature distribution along wellbore. When the eccentricities for the vertical, inclined, and horizontal sections were set to e = 0.6, e = 0.62, and e = 1, respectively, and the magnitude of revolutions per minute (RPM) of drill pipes was 100 r/min, the temperature difference at the wellbore bottom was only approximately 0.2 °C, which was negligible.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

250

500

750

1000

1250

1500

1750

2000

Flow Rate(m3/s)

Pre

ssur

e G

radi

ent (

Pa/

m)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

200

300

400

500

600

700

800

Flow Rate(m3/s)

Pre

ssur

e G

radi

ent(P

a/m

)

concentric(=0)Haciislamoglu(ecc, =0.6)Ahmed(ecc+rot, =0.6, RPM=60 r/min)Cartalos(ecc+rot, =0.6, RPM=60 r/min)

15 20 25 30 35 40 45 50 550

500

1000

1500

2000

2500

3000

Temperature( )℃

Mea

sure

d D

epth

(m)

WithoutPressure Drop

WithPressure Drop


15 20 25 30 35 40 45 50 550

500

1000

1500

2000

2500

3000

Temperature( )℃

Mea

sure

d D

epth

(m)

WithoutBit Pressure Drop

WithBit Pressure Drop


(a) (b)

(a) (b)

Figure 5. (a) Influence of pressure drops of the whole wellbore on temperature profile (b) Influence ofpressure drop of drill bit on temperature profile.

According to Figures 6 and 7, the eccentricity and rotation of drill pipes had insignificant effectson the temperature distribution along wellbore. When the eccentricities for the vertical, inclined,and horizontal sections were set to e = 0.6, e = 0.62, and e = 1, respectively, and the magnitude ofrevolutions per minute (RPM) of drill pipes was 100 r/min, the temperature difference at the wellborebottom was only approximately 0.2 ◦C, which was negligible.

Energies 2018, 11, 2414 16 of 28


Figure 6. Comparison between temperature profiles with eccentric drill pipes and with concentric drill pipes (Eccentricities: e = 0.6 for vertical section, e = 0.62 for inclined section and e = 1 for horizontal section).

Figure 7. Comparison between temperature profiles under eccentric conditions in drill pipes with and without rotations (RPM = 100 r/min).

3.2. Influence of Nu with Taylor Vortex under Various Rotation Rates of Drill Pipe

For the annulus in which the inner wall was rotary, the Taylor vortex occurred under a certain condition; a phenomenon that had been validated by numerous studies [40,42,45,46]. In this regard, the drilling fluid flows in the annulus between the wellbore wall and drill pipe were no exception [32]. The appearance of Taylor vortices certainly made a difference on the heat transfer in the annulus. Thus, the consideration of the effect of the Taylor vortex was important in predicting the temperature profile during drilling procedures. In order to analyze the convective heat transfer coefficient, α = λeNu/Dh, an analysis of the variation of Nu was required while the Nu value was affected by the appearance of the Taylor vortex. In previous studies, the drilling fluid flow in the annulus was assumed to be laminar. However, with respect to the effect of drill pipe rotation, it was found that it triggered Taylor vortices under certain rotation rates of drill pipes. In this study, it was maintained that the flow regime in annulus with rotatory inner drill pipes was of the laminar + Taylor vortices mode instead of the pure laminar flow regime. Thus, the effect of the Taylor vortex on Nu and temperature distribution was analyzed. In this regard, we mainly used the expression of Simmers and Coney [48] to conduct our analysis.

15 20 25 30 35 40 45 50 550

500

1000

1500

2000

2500

3000

Mea

sure

d D

epth

(m)

Temperature( )℃


41 42 43 44 452400

2600

2800

3000

Concentric

Eccentric

15 20 25 30 35 40 45 50 550

500

1000

1500

2000

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Figure 6. Comparison between temperature profiles with eccentric drill pipes and with concentric drillpipes (Eccentricities: e = 0.6 for vertical section, e = 0.62 for inclined section and e = 1 for horizontal section).


Figure 6. Comparison between temperature profiles with eccentric drill pipes and with concentric drill pipes (Eccentricities: e = 0.6 for vertical section, e = 0.62 for inclined section and e = 1 for horizontal section).

Figure 7. Comparison between temperature profiles under eccentric conditions in drill pipes with and without rotations (RPM = 100 r/min).


For the annulus in which the inner wall was rotary, the Taylor vortex occurred under a certain condition; a phenomenon that had been validated by numerous studies [40,42,45,46]. In this regard, the drilling fluid flows in the annulus between the wellbore wall and drill pipe were no exception [32]. The appearance of Taylor vortices certainly made a difference on the heat transfer in the annulus. Thus, the consideration of the effect of the Taylor vortex was important in predicting the temperature profile during drilling procedures. In order to analyze the convective heat transfer coefficient, α = λeNu/Dh, an analysis of the variation of Nu was required while the Nu value was affected by the appearance of the Taylor vortex. In previous studies, the drilling fluid flow in the annulus was assumed to be laminar. However, with respect to the effect of drill pipe rotation, it was found that it triggered Taylor vortices under certain rotation rates of drill pipes. In this study, it was maintained that the flow regime in annulus with rotatory inner drill pipes was of the laminar + Taylor vortices mode instead of the pure laminar flow regime. Thus, the effect of the Taylor vortex on Nu and temperature distribution was analyzed. In this regard, we mainly used the expression of Simmers and Coney [48] to conduct our analysis.

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Figure 7. Comparison between temperature profiles under eccentric conditions in drill pipes with andwithout rotations (RPM = 100 r/min).


For the annulus in which the inner wall was rotary, the Taylor vortex occurred under a certaincondition; a phenomenon that had been validated by numerous studies [40,42,45,46]. In this regard,the drilling fluid flows in the annulus between the wellbore wall and drill pipe were no exception [32].The appearance of Taylor vortices certainly made a difference on the heat transfer in the annulus. Thus,the consideration of the effect of the Taylor vortex was important in predicting the temperature profileduring drilling procedures. In order to analyze the convective heat transfer coefficient, α = λeNu/Dh,an analysis of the variation of Nu was required while the Nu value was affected by the appearanceof the Taylor vortex. In previous studies, the drilling fluid flow in the annulus was assumed to belaminar. However, with respect to the effect of drill pipe rotation, it was found that it triggered Taylorvortices under certain rotation rates of drill pipes. In this study, it was maintained that the flow regimein annulus with rotatory inner drill pipes was of the laminar + Taylor vortices mode instead of thepure laminar flow regime. Thus, the effect of the Taylor vortex on Nu and temperature distributionwas analyzed. In this regard, we mainly used the expression of Simmers and Coney [48] to conductour analysis.

Under the regime of laminar + Taylor vortices and axial flow rate unchanged, it can be observedin Figure 8 that the heat transfer in the wellbore was enhanced compared to that under the purelaminar regime. This was because under the laminar + Taylor vortices state, the temperature in the

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drill pipe was greater than that without Taylor vortices. Moreover, when Taylor vortices occurred,the temperature in the annulus at the upper wellbore section, where the temperature was higher thanthe geothermal temperature, was lower than that under the pure laminar state.


Under the regime of laminar + Taylor vortices and axial flow rate unchanged, it can be observed in Figure 8 that the heat transfer in the wellbore was enhanced compared to that under the pure laminar regime. This was because under the laminar + Taylor vortices state, the temperature in the drill pipe was greater than that without Taylor vortices. Moreover, when Taylor vortices occurred, the temperature in the annulus at the upper wellbore section, where the temperature was higher than the geothermal temperature, was lower than that under the pure laminar state.

Figure 8. Temperature profile considering the effect of drill pipe rotation. (RPM = 100 r/min).

However, at the lower wellbore section, where the annulus temperature was lower than the geothermal temperature, the temperature was higher when Taylor vortices occurred. Figure 9a shows that when considering the Taylor vortex, the wellbore bottom temperature was always greater than that under the pure laminar state, and the expression provided by Seder and Tate [34] was used for the pure laminar regime to estimate the wellbore bottom temperature. The temperature remained unchanged when the rotation rate varied because Nu was merely influenced by the axial velocity. On the other hand, the bottom temperature increased as the rotation rate increased when the Taylor vortex emerged. This was because Nu was simultaneously affected by the rotation and axial flow rates. As seen in Figure 9b, the variation of the wellbore bottom temperature as Nu increased under the laminar + Taylor vortices regime was similar with the varying trend of the temperature in relation to the rotation rate shown in Figure 8a, where the wellbore bottom temperature increased as Nu increased. The phenomena revealed that Taylor vortices considerably intensified the heat transfer in the wellbore.

Figure 9. Variation of temperature at wellbore bottom with the increase in (a) rotating rate and (b) Nusselt number, Nu.

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However, at the lower wellbore section, where the annulus temperature was lower than thegeothermal temperature, the temperature was higher when Taylor vortices occurred. Figure 9a showsthat when considering the Taylor vortex, the wellbore bottom temperature was always greater thanthat under the pure laminar state, and the expression provided by Seder and Tate [34] was used forthe pure laminar regime to estimate the wellbore bottom temperature. The temperature remainedunchanged when the rotation rate varied because Nu was merely influenced by the axial velocity.On the other hand, the bottom temperature increased as the rotation rate increased when the Taylorvortex emerged. This was because Nu was simultaneously affected by the rotation and axial flow rates.As seen in Figure 9b, the variation of the wellbore bottom temperature as Nu increased under thelaminar + Taylor vortices regime was similar with the varying trend of the temperature in relation tothe rotation rate shown in Figure 8a, where the wellbore bottom temperature increased as Nu increased.The phenomena revealed that Taylor vortices considerably intensified the heat transfer in the wellbore.


Under the regime of laminar + Taylor vortices and axial flow rate unchanged, it can be observed in Figure 8 that the heat transfer in the wellbore was enhanced compared to that under the pure laminar regime. This was because under the laminar + Taylor vortices state, the temperature in the drill pipe was greater than that without Taylor vortices. Moreover, when Taylor vortices occurred, the temperature in the annulus at the upper wellbore section, where the temperature was higher than the geothermal temperature, was lower than that under the pure laminar state.


However, at the lower wellbore section, where the annulus temperature was lower than the geothermal temperature, the temperature was higher when Taylor vortices occurred. Figure 9a shows that when considering the Taylor vortex, the wellbore bottom temperature was always greater than that under the pure laminar state, and the expression provided by Seder and Tate [34] was used for the pure laminar regime to estimate the wellbore bottom temperature. The temperature remained unchanged when the rotation rate varied because Nu was merely influenced by the axial velocity. On the other hand, the bottom temperature increased as the rotation rate increased when the Taylor vortex emerged. This was because Nu was simultaneously affected by the rotation and axial flow rates. As seen in Figure 9b, the variation of the wellbore bottom temperature as Nu increased under the laminar + Taylor vortices regime was similar with the varying trend of the temperature in relation to the rotation rate shown in Figure 8a, where the wellbore bottom temperature increased as Nu increased. The phenomena revealed that Taylor vortices considerably intensified the heat transfer in the wellbore.

Figure 9. Variation of temperature at wellbore bottom with the increase in (a) rotating rate and (b) Nusselt number, Nu.

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Figure 9. Variation of temperature at wellbore bottom with the increase in (a) rotating rate and(b) Nusselt number, Nu.

3.3. The Influence of Mechanical Friction under Various Rotation Rates of the Drill Pipe

In the drilling procedure, the drill pipes always had to rotate at high speeds in order to drivethe drill bit to break the rocks. Hence, continuous contact and friction between the rock and drill bit,

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as well as with the drill pipes was inevitable. Therefore, as a result of mechanical friction, a substantialamount of heat was produced and the drilling fluid was gradually heated. In this study, we firstlycomputed the torque distribution (shown in Figure 10) along all drill pipes and drill bit according tostudies conducted by Aadnøy and Andersen [53], and Warrant [52], and thereafter obtained the heatquantity in each small segment.


3.3. The Influence of Mechanical Friction under Various Rotation Rates of the Drill Pipe

In the drilling procedure, the drill pipes always had to rotate at high speeds in order to drive the drill bit to break the rocks. Hence, continuous contact and friction between the rock and drill bit, as well as with the drill pipes was inevitable. Therefore, as a result of mechanical friction, a substantial amount of heat was produced and the drilling fluid was gradually heated. In this study, we firstly computed the torque distribution (shown in Figure 10) along all drill pipes and drill bit according to studies conducted by Aadnøy and Andersen [53], and Warrant [52], and thereafter obtained the heat quantity in each small segment.

Figure 10. Distribution of torque along wellbore measured depth (RPM = 100 r/min).

Evidently, the heat quantity was from the inner heat source because of mechanical friction. In the discussion in this section, it was assumed that all the heat generated by mechanical friction at the drilling pipes and drill bit were absorbed by the drilling fluid in the annulus.

The wellbore temperature distribution in the annulus under pure laminar conditions is plotted in Figure 11. It can be seen from Figure 11a that the temperature in the drill pipe and annulus was greater when mechanical frictions of the drill bit and drill pipes were considered than when they were neglected. As the depth of the well increased, the temperature difference between the two conditions increased and peaked at the wellbore bottom. In this calculation example, when the rotation speed was 100 r/min, the annulus temperature difference with and without mechanical friction at the bottom of the well was 8.4 °C. When friction was not considered, the bottom temperature increased by 20.26%. Therefore, mechanical frictions along the wellbore had a considerable influence on the temperature at the wellbore bottom. When the effects of frictional heat at the drill bit were separately investigated, it can be seen from Figure 11b that when only the mechanical friction on the outer wall of drill pipes was considered, the wellbore bottom temperature was approximately 2 °C lower than the temperature at which the drill bit friction was considered. Thus, under this condition, the change in the bottom hole temperature caused by friction at the drill bit was approximately 25.5% of the total temperature difference. In addition, it can be observed from Figure 11b that when the frictional heat source at the drill bit was considered, there was a sudden increase in temperature as the drilling fluid flowed into the annulus from the drill pipes at the wellbore bottom, and that this temperature variation only affected a limited distance above the drill bit. Evidently, the temperature distribution was not affected by heat sources produced at the drill bit beyond a certain distance above it. With reference to Figures 5b and 11b, if the heat source at the drill bit was not taken into consideration, then, there was a temperature increase as the fluid flowed upwards through the annulus. The highest temperature in the annulus occurred at a certain distance from the wellbore bottom. However, when the influence of the heat source at the drill bit was considered, the

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Figure 10. Distribution of torque along wellbore measured depth (RPM = 100 r/min).

Evidently, the heat quantity was from the inner heat source because of mechanical friction. In thediscussion in this section, it was assumed that all the heat generated by mechanical friction at thedrilling pipes and drill bit were absorbed by the drilling fluid in the annulus.

The wellbore temperature distribution in the annulus under pure laminar conditions is plottedin Figure 11. It can be seen from Figure 11a that the temperature in the drill pipe and annulus wasgreater when mechanical frictions of the drill bit and drill pipes were considered than when they wereneglected. As the depth of the well increased, the temperature difference between the two conditionsincreased and peaked at the wellbore bottom. In this calculation example, when the rotation speedwas 100 r/min, the annulus temperature difference with and without mechanical friction at the bottomof the well was 8.4 ◦C. When friction was not considered, the bottom temperature increased by 20.26%.Therefore, mechanical frictions along the wellbore had a considerable influence on the temperature atthe wellbore bottom. When the effects of frictional heat at the drill bit were separately investigated,it can be seen from Figure 11b that when only the mechanical friction on the outer wall of drill pipeswas considered, the wellbore bottom temperature was approximately 2 ◦C lower than the temperatureat which the drill bit friction was considered. Thus, under this condition, the change in the bottomhole temperature caused by friction at the drill bit was approximately 25.5% of the total temperaturedifference. In addition, it can be observed from Figure 11b that when the frictional heat source atthe drill bit was considered, there was a sudden increase in temperature as the drilling fluid flowedinto the annulus from the drill pipes at the wellbore bottom, and that this temperature variationonly affected a limited distance above the drill bit. Evidently, the temperature distribution was notaffected by heat sources produced at the drill bit beyond a certain distance above it. With reference toFigures 5b and 11b, if the heat source at the drill bit was not taken into consideration, then, there was atemperature increase as the fluid flowed upwards through the annulus. The highest temperature inthe annulus occurred at a certain distance from the wellbore bottom. However, when the influence ofthe heat source at the drill bit was considered, the phenomenon of temperature inversion mentionedby Kabir and Hasan [7] reduced or completely disappeared. A noteworthy difference between thetemperature profile in the vertical well and bottom hole of the horizontal well was that when the heat

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source term was taken into account, the temperature in the latter may be greater than the originalgeothermal temperature. It can be seen in Figure 11 that, overall, the annulus temperature was higherthan the original formation temperature when frictional heat sources were included.


phenomenon of temperature inversion mentioned by Kabir and Hasan [7] reduced or completely disappeared. A noteworthy difference between the temperature profile in the vertical well and bottom hole of the horizontal well was that when the heat source term was taken into account, the temperature in the latter may be greater than the original geothermal temperature. It can be seen in Figure 11 that, overall, the annulus temperature was higher than the original formation temperature when frictional heat sources were included.

Figure 11. Wellbore temperature profile (a) with and without total mechanical friction (b) with and without considering the effect of mechanical friction on the drill bit (RPM = 100 r/min).

The rotation rate of drill pipes affected the amount of heat generated during drilling processes. In Figure 12, without considering the effect of rotation on the convection heat transfer, the red line with circular markers and blue line with triangular markers show the temperature variations at the wellbore bottom when the heat source at the drill bit was taken into account and excluded, respectively. In both cases, it wa

heat transfer behaviors in horizontal wells considering ......energies 2018, 11, 2414 4 of 28 (4)...

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